Properties of the Beurling generalized primes

AbstractTextIn this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that limx→∞1lnx∏p⩽x(1−p−1)−1=Aeγ, where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit M=limx→∞(∑p⩽xp−1−ln(lnx)) exists. We also show that this limit coincides with limα→0+(∑pp−1(lnp)−α−1/α); for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that |N(x)−[x]|